Conservation Laws and Boundary Conditions

We begin by derving the equations of motion used to model ocean waves. The equations of motion of a general fluid are given by the celebrated Navier Stokes equations. However, for the large scale processes that occur in ocean waves many simplifications are possible.

Non-Linear Free-surface Condition

We begin by defining the coordinate system.

[math]\displaystyle{ \begin{matrix} &(X,Y,Z) &: &\mbox{Earth Fixed Coordinate System} \\ &\vec X &: &\mbox{Fixed Eulerian Vector} \\ &\vec V &: &\mbox{Flow Velocity Vector at} \ \vec X \\ &\zeta &: &\mbox{Free Surface Elevation} \end{matrix} }[/math]

At the moment we have not yet defined the region of space which is occupied by the fluid. However, the free surface plays a very important role in the propagation of ocean waves.

The most important assumption we make is that the fluid is an ideal fluid. This means that there are no shear stresses due to viscosity and that the flow is irrotational. This means that

[math]\displaystyle{ \nabla \times \vec V = 0 }[/math]

We now introduce the very important concept of the velocity potential. Essentially this allows us to express the solution as a function of a scalar (a function which has a single value as opposed to a vector function which has multiple values) rather than a vector through out the fluid domain. There is an important theorem in vector calculus [1] that if [math]\displaystyle{ \nabla \times \vec V = 0 }[/math] then we can express the irrotational vector as the gradiant of a scalar function, i.e.

[math]\displaystyle{ \vec V = \nabla \Phi }[/math]

where [math]\displaystyle{ \Phi(\vec{X},t) }[/math] is called the velocity potential.

It turns of that the potential flow model of surface wave propagation and wave-body interactions is very accurate for the kind of length and time scales which are important in the ocean. It is important to realise however that we have made considerably simplifications and that certain processes, most notably wave breaking are in no way covered by this theorey. In fact, the process of wave breaking is extremely complicated and is much less well understood that the potential flow model.

Conservation of mass

The key equation we will solve to understand ocean waves is Laplace's equation which is derived very simply from the expression for the velocity potential. We assume that the fluid is incompressible so that conservation of mass gives us the following condition

[math]\displaystyle{ \nabla \cdot \vec V = 0 }[/math]

This condition in turn implies, using the definition of the velocity potential that

[math]\displaystyle{ \nabla \cdot \nabla \Phi = 0 \Rightarrow \nabla^2 \Phi = 0 }[/math]

or

[math]\displaystyle{ \frac{\partial^2 \Phi}{\partial X^2} + \frac{\partial^2\Phi}{\partial Y^2} + \frac{\partial^2\Phi}{\partial Z^2} = 0, \quad \mbox{Laplace Equation} }[/math]

.

Conservation of linear momentum

We begin with Euler's equation in the absence of viscosity

[math]\displaystyle{ \frac{\partial \vec V}{\partial t} + (\vec V \cdot \nabla) \vec V = - \frac1{\rho} \nabla P + \vec g }[/math]

where [math]\displaystyle{ P(\vec X, t) }[/math] is the fluid Pressure at [math]\displaystyle{ (\vec X, t) }[/math] and [math]\displaystyle{ \vec g = - \vec k g }[/math] is the acceleration due to gravity where [math]\displaystyle{ \vec k }[/math] is the unit vector pointing in the positive z-direction (so we are now setting the [math]\displaystyle{ z }[/math] coordinate to point in the vertical direction. Finally [math]\displaystyle{ \rho }[/math] is the water density.

We then use the folliwng vector identity

[math]\displaystyle{ (\vec V \cdot \nabla) \vec V = \frac 1{2} \nabla (\vec V \cdot \vec V) - \vec V \times ( \nabla \times \vec V) }[/math]

and since we have irrotational flow (i.e. [math]\displaystyle{ \nabla \times \vec V = 0 }[/math]) Euler's equation becomes

[math]\displaystyle{ \frac{\partial \vec V}{\partial t} + \frac 1{2} \nabla (\vec V \cdot \vec V) = - \frac 1{\rho} \nabla P - \nabla (g Z) }[/math]

where we have used [math]\displaystyle{ \nabla Z = \vec K }[/math].

We now substitute [math]\displaystyle{ \vec V = \nabla \Phi }[/math] and we obtain

[math]\displaystyle{ \nabla (\frac{\partial \Phi}{\partial t} + \frac 1{2} \nabla \Phi \cdot \nabla \Phi + \frac {P}{e} + g Z ) = 0 }[/math]

We now observe that if

[math]\displaystyle{ \nabla F( \vec X, t) =0 \quad \Longrightarrow \quad F (\vec X, t) = C }[/math]

where [math]\displaystyle{ C }[/math] is an arbitrary constant.

Bernoulli's equation

Bernoulli's equation follows

[math]\displaystyle{ \frac{\partial \Phi}{\partial t} + \frac 1{2} \nabla \Phi \cdot \nabla \Phi + \frac {P}{\rho} + g Z = C }[/math]

or

[math]\displaystyle{ \frac{P}{\rho} = - \frac{\partial\Phi}{\partial t}-\frac{1}{2}\nabla\Phi \cdot \nabla \Phi - g Z + C }[/math]

The value of the constant [math]\displaystyle{ C }[/math] is immaterial. It is also woth noting that the angular momentum conservation principle is contained in [math]\displaystyle{ \nabla \times \vec V = 0 }[/math] In particualr, if the particles are modelled as spheres, this equation implies no angular velocity at all times.

Derivation of Nonlinear Free-surface Condition

A very important result is the boundary condition at the free surface of the fluid and air. We will present two methods to derive this result.

Method I

On [math]\displaystyle{ Z=\zeta; \ P=P_a \equiv \mbox{Atmospheric Pressure} }[/math]

From Bernoulli:

[math]\displaystyle{ \longrightarrow \frac{P_a}{\rho}=-\frac{\partial\Phi}{\partial t}-\frac{1}{2}\nabla\Phi\cdot\nabla\Phi-g\zeta+\mathbb{C} \qquad \mbox{on} \ Z=\zeta(X,Y,t) }[/math]

On [math]\displaystyle{ Z=\zeta }[/math] The mathematical function

[math]\displaystyle{ Z-\zeta(X,Y,t)\equiv\tilde{f}(X,Y,Z,t) }[/math]

is always zero when tracing a fluid particle on the free surface. So the substantial or total derivative of [math]\displaystyle{ \tilde{f} }[/math] must vanish, thus

[math]\displaystyle{ \frac{D\tilde{f}}{Dt}=0=\left ( \frac{\partial}{\partial t} + \vec V \cdot \nabla \right ) \tilde{f}=0, \qquad \mbox{on} \ Z=\zeta }[/math]

Expanding we obtain:

[math]\displaystyle{ \left (\frac{\partial}{\partial t} + \vec V \cdot \nabla \right ) (Z-\zeta) =0, \qquad \mbox{on} \ Z=\zeta }[/math]

[math]\displaystyle{ \longrightarrow \frac{\partial\zeta}{\partial t}+\frac{\partial\Phi}{\partial X} \frac{\partial\zeta}{\partial X}+\frac{\partial\Phi}{\partial Y}\frac{\partial\zeta}{\partial Y}=\frac{\partial\Phi}{\partial Z}, \ Z=\zeta \qquad \mbox{(Kinematic free-surface condition)} }[/math]

From Bernoulli's equation we obtain the dynamic free surface condition:

[math]\displaystyle{ \longrightarrow \frac{\partial\Phi}{\partial t} + \frac{1}{2} \nabla \Phi \cdot \nabla \Phi + g \zeta = \mathbb{C} - \frac{P_a}{\rho}, \ Z=\zeta \qquad \mbox{(Dynamic free-surface condition)} }[/math]

Constants in Bernoulli's equation may be set equal to zero when we are eventually interested in integrating pressures over closed or open boundaries (floating or submerged bodies) to obtain forces & moments. This follows from a simple application of one of the two gauss vector theorems we will use a lot in this course:

Guass I: [math]\displaystyle{ \vec n: \ \mbox{unit normal vector pointing inside the volume} \ \bar{V} }[/math]
[math]\displaystyle{ f(\bar{X}: \ \mbox{arbitrary sufficiently differentiable scalar function} }[/math]
[math]\displaystyle{ \mbox{vector identity} \ \iiint \nabla f dv = -\iint f \vec n ds }[/math]

Note the three scalar identities that follow:

[math]\displaystyle{ \iiint_{\bar{V}} \frac{\partial f}{\partial x} dv = - \iint_{s} f n_1 ds }[/math]

[math]\displaystyle{ \iiint_{bar{v}} \frac{\partial f}{\partial y} dv = - \iint_{s} f n_2 ds }[/math]

[math]\displaystyle{ \iiint_{bar{v}} \frac{\partial f}{\partial z} dv = - \iint_{s} f n_3 ds }[/math]

Guass II: [math]\displaystyle{ \vec V: \ \mbox{arbitrary sufficiently differentiable vector function} }[/math]

Scalar identity: [math]\displaystyle{ \iiint_{\bar{V}} \nabla \cdot \vec V = - \iint_{s} \vec V \cdot \vec n ds }[/math]

Scalar identity ofter used to prove mass conservation principle.

Definition of force & moment in terms of fluid pressure.

It follows from Gauss I that if [math]\displaystyle{ \rho = mathbb{C} }[/math] the force and moment over a closed boundary S vanish identically. Hence without loss of generality in the context of wave body interactions we will set [math]\displaystyle{ mathbb{C}=0 }[/math]. It follows that the dynamic free surface condition takes the form

[math]\displaystyle{ \zeta (x,y,t) = - \frac{1}{g} \left \{ \frac{\partial\Phi}{\partial t} + \frac{1}{2} \nabla\Phi \cdot \nabla\Phi \right \}, \qquad Z=\zeta }[/math]

Method II

When tracing a fluid particle on the free surface the hydrodynamic pressure given by Bernoulli (after the constant [math]\displaystyle{ \mathbb{C} }[/math] has been set equal to zero) must vanish as we follow the particle:

[math]\displaystyle{ \frac{D}{Dt} \left \{ \frac{\partial\Phi}{\partial t} + \frac{1}{2} \nabla\Phi \cdot \nabla\Phi+gZ \right \} =0, \qquad Z=\zeta }[/math]

or

[math]\displaystyle{ \left ( \frac{\partial}{\partial t} + \vec V \cdot \nabla \right ) \left ( \frac{\partial\Phi}{\partial t} + \frac{1}{2} + \frac{1}{2} \nabla\Phi \cdot \nabla\Phi +gZ \right ) =0, \qquad Z=\zeta }[/math]

This condition also follows upon elimination of [math]\displaystyle{ \zeta }[/math] from the kinematic & dynamic conditions derived under method I.

This completes the statement of the nonlinear boundary value problem satisfied by surface waves of large amplitude in potential flow and in the absence of wave breaking.

Ocean Wave Interaction with Ships and Offshore Energy Systems